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Theorem 2rmorex 3308
Description: Double restricted quantification with "at most one," analogous to 2moex 2374. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2rmorex  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A  ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem 2rmorex
StepHypRef Expression
1 nfcv 2629 . . 3  |-  F/_ y A
2 nfre1 2925 . . 3  |-  F/ y E. y  e.  B  ph
31, 2nfrmo 3037 . 2  |-  F/ y E* x  e.  A  E. y  e.  B  ph
4 rspe 2922 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
54ex 434 . . . . 5  |-  ( y  e.  B  ->  ( ph  ->  E. y  e.  B  ph ) )
65ralrimivw 2879 . . . 4  |-  ( y  e.  B  ->  A. x  e.  A  ( ph  ->  E. y  e.  B  ph ) )
7 rmoim 3303 . . . 4  |-  ( A. x  e.  A  ( ph  ->  E. y  e.  B  ph )  ->  ( E* x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A  ph ) )
86, 7syl 16 . . 3  |-  ( y  e.  B  ->  ( E* x  e.  A  E. y  e.  B  ph 
->  E* x  e.  A  ph ) )
98com12 31 . 2  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  ( y  e.  B  ->  E* x  e.  A  ph ) )
103, 9ralrimi 2864 1  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   E.wrex 2815   E*wrmo 2817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rmo 2822
This theorem is referenced by:  2reu2  31687
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